• Mixed Numbers: An Educator’s Guide

  • In many countries mixed numbers are introduced in middle primary. But why? And what use are they?

    Why and where?

    Whether  you teach in Australia, the United States, Canada, New Zealand, United  Kingdom, Hong Kong or Singapore; mixed numbers are introduced in middle  primary. But why? And what use are they?

    Those  familiar with more senior mathematics would know that we rarely use  mixed numbers in those settings, so why include mixed numbers in the  junior curriculum?

    The detail that each curriculum authority provides about mixed numbers is usually sparse.

    For  example in Australia, mixed numbers are mentioned in Year 4, here is  the only mention of mixed numbers in all of primary school fractional

    from here

    In the Common Core in the United States, it is only mentioned through one of the examples provided for 4.NF.3b.

    from here

    Singapore details the most, in Primary Year 4 it lists 5 dot points  about how to relate mixed numbers and improper fractions together.

    How I  interpret this, is not that mixed numbers are being considered  unimportant, but instead that they actually interconnect with broader  fraction work, utilising both representations of mixed numbers and  improper fractions where it makes sense to do so. Hence developing the  number sense surrounding mixed numbers throughout fractional  development.

    The conceptual process for developing understanding of mixed numbers

    Having  worked with programs focussed on the development of deep understanding  of fractions for over a decade, I have come to see that the following  presents as a good foundation for mixed numbers.

    1. Develop  the idea of a mixed number as being another representation of the  improper equivalent. This can be developed through the use of diagrams  and manipulatives.

    Formal conversion rules, such as multiplying and adding should not be  taught in the introduction, but students should be allowed to develop  this understanding through playing with the manipulatives, conversation  and explanation of their thinking.

    leave these formal rules out for now, concentrate on developing fraction sense

    2. Plot and identify the size of mixed numbers on numbers lines, or with other physical representations

    3. Order and count with mixed numbers. Counting both using mixed numbers,

    and with improper fractions,

    4. Reason and understand the mixed number representation from an  improper fraction, for example that 6 fifths is 1 wholes (5 fifths) and 1  more fifth. Not by way of conversion, but by way of understanding. Use  of pictorial representations is essential here.

    5. Benchmarking, (or rounding) mixed numbers, to common values like 0, ½, 1, or any whole number.

    6a. Then, when the curriculum moves on to introduce addition of  fractions, mixed numbers are integrated here. Starting with recognising  that a mixed number is already an addition of sorts, that 1⅕ is 1 whole  and 1 more fifth.

    Addition is scaffolded well, starting from converting final answers between improper and mixed numbers.

    2 thirds + 5 thirds is 7 thirds,

    which is 7/3

    which is 2 wholes and 1 more thirds.

    The  integration of the conversion from improper to mixed at this stage is  essential for developing strong number sense around the size of the  number 7/3. We can see from its mixed number version that it is larger  than 2 wholes, but less than 3 wholes.

    A more advanced number sense idea here, would be that it is bigger than 2 wholes but smaller than 2 ½.

    6b.  We now introduce the idea of adding and subtracting mixed numbers. We  do this in a similar way to how we develop decimal addition and  subtraction that usually comes a few years later, and that is through  the understanding that 2 1/4 + 4 1/4 is pictured…

    Some students can at this stage connect 2/4 to be the same as ½.

    Generally  the approach to elicit understanding here is to consider the  whole-number parts first, followed by the fractional parts. This  approach also yields a good estimation prior to the calculation. 2+ 4 is  6, so we must be more than 6. Both fractional components are less than ½  so the sum must be less than 7.

    Resist  here the procedural approach to convert both fractions to improper and  then add them, it detracts from the intuitive nature of what is  occurring. The convert first approach should be something that students  recognise as efficient when adding or subtracting more complicated  examples such as

    Although it could be argued here that a reasonable partition could be made,

    None of which is nicely formatted for a “pretty written answer”, but the  reasoning is deep and connects all the ideas and strategies of whole  number partitioning to fraction computation.

    Why do we stop using mixed numbers?

    Those  that are familiar with higher level mathematics, would know that beyond  this introduction in the early/middle years that mixed numbers are  rarely if ever used in formal mathematics. Mostly improper fractions are  used and kept for calculations. Why is this?

    Well  one reason is that in younger grades we spend far more time adding and  subtracting, and mixed number representations can be quite useful here.  i.e. Adding and subtracting whole number components first then adding  and subtracting the fractional components.

    They  are also useful in developing the number sense associated with the  different representations. For example, 1⅖ and 7/5. The 1⅖ immediately  recognisable in size as something between 1 and 2. The 7/5 immediately  recognisable as greater than 1 whole, and something that is made up of 7  equal size parts. It is also easier to see from the mixed number  representation of 1⅖ that it is less than 1½. So an estimate for a  computation using addition and subtraction using mixed numbers can in  some ways be easier to arrive at.

    In  latter years we do more with multiplication and division, and we rarely  keep mixed number representations in these types of calculations, in  fact algorithmically, we convert mixed numbers to improper at this stage  and then multiply, or invert then multiply as needed. So it isn’t  necessarily that we stop using mixed numbers, more that our usage of  them becomes less important in our formal mathematical studies.

    Final thoughts

    One  thing that might be missing here is the bridge between what is  perceived as primary usage, and secondary non-usage. Maybe we need to  have students explore this deeply to understand how one representation  is more useful in some situations than others? Then this gap in  understanding about why we seem to use mixed numbers in earlier years  and not use mixed numbers in older years might not exist.

    So  mixed numbers have their place, it’s great to develop that sense of  size and we do need to be able to convert them for computation at  different points in study. Some argue that mixed numbers are more real  life, for example in recipes and in dividing physical quantities between  people. So maybe mixed numbers have greater importance in a numeracy  perspective, and improper have greater importance in formal mathematical  calculations?

    Either  way, curricula from around world follow a very similar pattern in the  development of mixed numbers and we all do it at about the same time in  schools. What we can benefit from here is understanding how the  importance of exploration, play and conversation can be in drawing out  the conceptual understanding of what mixed numbers are, and in relating  the relevance of mixed numbers to the context, the calculation or in  estimation and assessing reasonablness.

    Thanks to Andie Hoyt.